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Conformal Graph Prediction with Z-Gromov Wasserstein Distances

Gabriel Melo, Thibaut de Saivre, Anna Calissano, Florence d'Alché-Buc

TL;DR

This work proposes a conformal prediction framework for graph-valued outputs, providing distribution--free coverage guarantees in structured output spaces, and introduces Score Conformalized Quantile Regression (SCQR), an extension of Conformalized Quantile Regression to handle complex output spaces such as graph--valued outputs.

Abstract

Supervised graph prediction addresses regression problems where the outputs are structured graphs. Although several approaches exist for graph-valued prediction, principled uncertainty quantification remains limited. We propose a conformal prediction framework for graph-valued outputs, providing distribution-free coverage guarantees in structured output spaces. Our method defines nonconformity via the Z-Gromov-Wasserstein distance, instantiated in practice through Fused Gromov-Wasserstein (FGW), enabling permutation invariant comparison between predicted and candidate graphs. To obtain adaptive prediction sets, we introduce Score Conformalized Quantile Regression (SCQR), an extension of Conformalized Quantile Regression (CQR) to handle complex output spaces such as graph-valued outputs. We evaluate the proposed approach on a synthetic task and a real problem of molecule identification.

Conformal Graph Prediction with Z-Gromov Wasserstein Distances

TL;DR

This work proposes a conformal prediction framework for graph-valued outputs, providing distribution--free coverage guarantees in structured output spaces, and introduces Score Conformalized Quantile Regression (SCQR), an extension of Conformalized Quantile Regression to handle complex output spaces such as graph--valued outputs.

Abstract

Supervised graph prediction addresses regression problems where the outputs are structured graphs. Although several approaches exist for graph-valued prediction, principled uncertainty quantification remains limited. We propose a conformal prediction framework for graph-valued outputs, providing distribution-free coverage guarantees in structured output spaces. Our method defines nonconformity via the Z-Gromov-Wasserstein distance, instantiated in practice through Fused Gromov-Wasserstein (FGW), enabling permutation invariant comparison between predicted and candidate graphs. To obtain adaptive prediction sets, we introduce Score Conformalized Quantile Regression (SCQR), an extension of Conformalized Quantile Regression (CQR) to handle complex output spaces such as graph-valued outputs. We evaluate the proposed approach on a synthetic task and a real problem of molecule identification.
Paper Structure (56 sections, 6 theorems, 39 equations, 5 figures, 9 tables)

This paper contains 56 sections, 6 theorems, 39 equations, 5 figures, 9 tables.

Table of Contents

  1. Introduction
  2. Related Work
  3. Prediction Models for Graphs.
  4. Conformal Prediction for Graphs.
  5. Background
  6. Notation
  7. Conformal Prediction
  8. Conformalized Quantile Regression
  9. Z-Gromov Wasserstein Distance
  10. Z-Gromov--Wasserstein distance.
  11. Weak isomorphism and metricity.
  12. Conformal Graph Prediction
  13. Problem Setup
  14. Graphs as $Z$-networks.
  15. Discrete $Z$-Gromov--Wasserstein for graphs.
  16. ...and 41 more sections

Key Result

Theorem 3.5

For any separable metric space $(\mathcal{Z}, d_\mathcal{Z})$ and $p \ge 1$, the distance $\mathrm{GW}^Z_p$ induces a genuine metric on $\mathcal{M}$.

Figures (5)

  • Figure 1: Conformal metabolite prediction set at $90\%$ marginal coverage for molecule identification on MassSpecGym. The green check-mark denotes the ground-truth.
  • Figure 2: Z-GW distance
  • Figure 3: Conformal Set Size Histogram (MassSpecGym).
  • Figure 4: Conformal coverage and set size behavior across tasks. Histograms show empirical coverage versus candidate set size. Scatter plots show conformal set size versus candidate set size, the black dashed $y=x$ line indicates no reduction, and points below the line demonstrate cases where conformal filtering reduces the set. Blue (resp. red) points indicate conformal sets containing the ground truth (resp. violations). For SCQR, the attribute function $\omega(x)$ is task-dependent: $\omega(x)=|\mathcal{L}(x)|$ (candidate set size) for Coloring, and $\omega(x)=\textsc{DreaMS}(x)$ (mass-spectrum embedding) for Metabolite.
  • Figure 5: Coloring dataset examples using Any2Graph as graph predictor.

Theorems & Definitions (25)

  • Definition 3.1: Exchangeability
  • Definition 3.2: Metric Measure Spaces
  • Definition 3.3: $Z$-networks
  • Definition 3.4: Weak isomorphism bauer2024z
  • Theorem 3.5: bauer2024z
  • Remark 3.6: Working with representatives in practice
  • Example 4.1: Fused Network Gromov-Wasserstein Distance
  • Example 4.2: Fused Gromov-Wassertein Distance vayer2020fused
  • Example 4.3: Gromov-Wasserstein Distance memoli2011gromov
  • Proposition 4.4: Weak isomorphism and permutation invariance
  • ...and 15 more